\chapter{Statistical Mechanics of Cooperation}
\label{chap:stat_mech}

\section{Hierarchical Hamiltonians}

We generalize classical spin systems to encode rule-dependent, level-specific couplings. Recalling \cref{def:hamiltonian}, we refine the coupling matrices to capture governance rules and structural heterogeneity.

\begin{definition}[Coupling tensors]
The intra-level tensor $\mathbf{J}^{(\ell)}$ has entries $J^{(\ell)}_{ijk}$ mapping rules $(r_i, r_j)$ to interaction strengths, while cross-level tensor $\mathbf{K}^{(\ell,\ell+1)}$ captures supervisory influence. The effective field on site $i$ becomes
\begin{equation}
H_i = \sum_{j \in \mathcal{N}_i^{\text{intra}}} J^{(\ell)}_{ijr_i r_j} s_j + \sum_{j \in \mathcal{N}_i^{\text{inter}}} K^{(\ell,\ell')}_{ij} s_j + h_i.
\end{equation}
\end{definition}

\begin{assumption}[Coupling sparsity]
Interaction tensors are sparse and banded, enabling efficient coarse-graining and simulation.
\end{assumption}

\section{Partition Functions and Free Energy}

The partition function $Z$ encodes macroscopic observables. Mean-field approximation yields free energy density
\begin{equation}
\mathcal{F} = -\sum_{\ell} \frac{1}{2} J^{(\ell)}_{\text{eff}} m_\ell^2 - \sum_{\ell < \ell'} K^{(\ell,\ell')}_{\text{eff}} m_\ell m_{\ell'} - \sum_{\ell} h_\ell m_\ell + \frac{1}{\beta}\sum_{\ell} S(m_\ell)
\end{equation}
with $S(m) = \frac{1+m}{2}\log\frac{1+m}{2} + \frac{1-m}{2}\log\frac{1-m}{2}$.

\begin{lemma}[Self-consistency equations]
Stationary points satisfy
\begin{equation}
m_\ell = \tanh\Big(\beta (J^{(\ell)}_{\text{eff}} m_\ell + \sum_{\ell'\neq \ell} K^{(\ell,\ell')}_{\text{eff}} m_{\ell'} + h_\ell)\Big).
\end{equation}
\end{lemma}

\begin{proof}
Derivative $\partial \mathcal{F} / \partial m_\ell = 0$ yields the stated expression.
\end{proof}

\section{Phase Transitions and Critical Surfaces}

Critical behavior depends on the spectral properties of the coupling graph.

\begin{theorem}[Layered critical temperature]
\label{thm:hierarchical_critical_temp}
Let $\mathbf{M}$ be the block matrix with entries $M_{\ell \ell'} = \beta J^{(\ell)}_{\text{eff}} \delta_{\ell \ell'} + \beta K^{(\ell,\ell')}_{\text{eff}} (1-\delta_{\ell \ell'})$. The system undergoes a second-order phase transition when the largest eigenvalue of $\mathbf{M}$ equals one. The corresponding critical inverse temperature is $\beta_c = 1/\lambda_{\max}(\mathbf{J}_{\text{eff}})$.
\end{theorem}

\begin{proof}
Linearize the self-consistency equations around $m_\ell = 0$ to obtain $\mathbf{m} = \mathbf{M} \mathbf{m}$. Non-trivial solutions arise when $\det(\mathbf{I} - \mathbf{M}) = 0$.
\end{proof}

Critical surfaces partition the parameter space $(J^{(\ell)}_{\text{eff}}, K^{(\ell,\ell')}_{\text{eff}}, h_\ell)$ into phases with distinct order properties. Chapter~\ref{chap:experiments} details procedures for estimating these surfaces empirically.

\section{Renormalization and Scaling Laws}

Renormalization group (RG) analysis tracks how effective couplings evolve with coarse-graining.

\begin{definition}[Hierarchical RG transformation]
Define block-spin variables $S^{(b)}_{\ell}$ averaging $b^d$ sites per level. The RG map $\mathcal{R}_b$ transforms coupling tensors $(\mathbf{J}, \mathbf{K}) \mapsto (\mathbf{J}', \mathbf{K}')$ via partial trace operations.
\end{definition}

\begin{proposition}[Scaling relation]
Near criticality, order parameters obey
\begin{equation}
\langle |m_\ell| \rangle \sim (T_c - T)^{\beta_\ell}, \qquad \chi_\ell \sim |T - T_c|^{-\gamma_\ell},
\end{equation}
where $(\beta_\ell, \gamma_\ell)$ derive from RG fixed-point eigenvalues.
\end{proposition}

\begin{remark}
Cross-level couplings induce mixed universality classes; we catalogue feasible scenarios in \cref{tab:universality}.
\end{remark}

\begin{table}[H]
    \centering
    \caption{Representative universality classes for hierarchical cooperation}
    \label{tab:universality}
    \begin{tabular}{p{3.5cm}p{3cm}p{3cm}p{3cm}}
        \toprule
        Coupling regime & Primary driver & $\beta_\text{eff}$ & $\gamma_\text{eff}$ \\
        \midrule
        Strong intra-level, weak cross-level & Layer autonomy & Classical Ising & 7/4 \\
        Balanced couplings & Supervisor alignment & 0.40--0.45 & 1.4--1.6 \\
        Dominant cross-level & Governance enforced order & Mean-field 1/2 & 1 \\
        \bottomrule
    \end{tabular}
\end{table}

\section{Numerical Validation Strategy}

Monte Carlo sampling (Metropolis, Wolff) and variational approximations calibrate theoretical predictions. Simulation requirements include:
\begin{itemize}
    \item Parameter sweeps across $J^{(\ell)}$, $K^{(\ell,\ell')}$, noise strengths, and field biases.
    \item Finite-size scaling runs with lattice sizes $L^d$ where $L \in \{32, 64, 128\}$.
    \item Logging magnetization, susceptibility, Binder cumulants, and information-theoretic metrics for cross-validation.
\end{itemize}

Chapter~\ref{chap:simulation} operationalizes these requirements via configuration schemas and instrumentation pipelines.

